| L(s) = 1 | + (−1.05 − 2.68i)2-s + (−0.556 − 7.43i)3-s + (−0.238 + 0.221i)4-s + (6.01 − 4.09i)5-s + (−19.3 + 9.32i)6-s + (4.14 + 18.0i)7-s + (−19.9 − 9.60i)8-s + (−28.2 + 4.25i)9-s + (−17.3 − 11.8i)10-s + (21.5 + 3.24i)11-s + (1.78 + 1.65i)12-s + (−10.5 + 13.2i)13-s + (44.1 − 30.1i)14-s + (−33.8 − 42.3i)15-s + (−4.96 + 66.3i)16-s + (39.0 − 12.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.372 − 0.949i)2-s + (−0.107 − 1.43i)3-s + (−0.0298 + 0.0277i)4-s + (0.537 − 0.366i)5-s + (−1.31 + 0.634i)6-s + (0.223 + 0.974i)7-s + (−0.881 − 0.424i)8-s + (−1.04 + 0.157i)9-s + (−0.548 − 0.373i)10-s + (0.589 + 0.0888i)11-s + (0.0428 + 0.0397i)12-s + (−0.225 + 0.283i)13-s + (0.842 − 0.575i)14-s + (−0.581 − 0.729i)15-s + (−0.0776 + 1.03i)16-s + (0.556 − 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.195867 - 1.20319i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.195867 - 1.20319i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-4.14 - 18.0i)T \) |
| good | 2 | \( 1 + (1.05 + 2.68i)T + (-5.86 + 5.44i)T^{2} \) |
| 3 | \( 1 + (0.556 + 7.43i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (-6.01 + 4.09i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-21.5 - 3.24i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (10.5 - 13.2i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-39.0 + 12.0i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-46.3 + 80.2i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-4.74 - 1.46i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-49.3 + 216. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-130. - 226. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (311. + 288. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-129. - 62.3i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (260. - 125. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-196. - 499. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-358. + 332. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-376. - 256. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-481. - 447. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-242. - 420. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-119. - 524. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-130. + 332. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (163. - 283. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (637. + 798. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (465. - 70.1i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + 359.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.27541203727370183741489050612, −13.04431302005114873728611458281, −12.08674205903905241373416942566, −11.52370070579368339325347317115, −9.747983330355553702710750821818, −8.684839231267340758297097081413, −6.94898723787333037700175193628, −5.68272776554336713254409644568, −2.47286956116947823780959976606, −1.22223753490711421829053738706,
3.61685096352754956989021428486, 5.37332627241112789265406027652, 6.82591475094397799348521144764, 8.284191015382154707934840792322, 9.739387655459622319973466512160, 10.47688781518107778684091579074, 11.88597993490451532325712148889, 13.97961650293799364329363011307, 14.75891566011901967488433219409, 15.75327588335937116445795685434
